The problem of estimating the parameters of a sinusoidal signal has received considerable attention in the literature and in various prior art patents. Such a problem arises in diverse engineering situations such as carrier tracking for communications systems and the measurement of Doppler in position location, navigation, and radar systems.
A variety of techniques have been proposed in the literature to solve such problems including, to mention a few, the application of the Fast Fourier Transform (FFT), one and two-dimensional Kalman filters based on a linearized model, a modified Kalman filter that results in a phase locked loop, and a digital phase locked loop derived on the basis of linear stochastic optimization.
U.S. Pat. No. 4,686,532 of McAulay describes a system for determining the strength of distributed scatterers (say P in number) that are possibly close to the sonar or radar relative to the array dimension of size I. Assuming that P scatterers rediate signals at some of the L Doppler frequencies known in advance, the method uses a least squares approach to estimate the complex amplitude of these P scatterers at each of these L possible frequencies. It is also assumed that various attenuations, time delays and angles between the P scatterers and I receivers are available via some other means. Since, of course, the Dopplers are not known a-priori, the method would essentially segment the entire Doppler range into intervals of delta-f and measure the complex signal amplitude of each of these frequencies and at each of the I receivers by some prior known method. Essentially, the method obtains the knowledge of the spectral contents of the scatterers from the known spectral contents of the received signals. It does not address the problem of how to obtain the spectral contents of the received signal. This latter problem is the subject of the present invention. Moreover, the framework of the scheme is also very computationally intensive as it involves an exhaustive search in the Doppler domain. It also does not take into consideration the time-varying Doppler--a clear possibility in a near field situation. Thus, in this framework, a discrete scatterer with time-varying Doppler would be identified by L complex amplitudes that are functions of time rather than identifying it as a single sinusoidal signal with time-varying amplitude, phase and frequency as would be more desirable.
U.S. Pat. No. 4,533,918 of Virnot describes a device working with radar equipment to obtain the location of a target. This is achieved by a Kalman filter type of circuit that operates on the samples of range measurements from the ship to a plurality of known landmarks. The range measurement is done by conventional pulse radar. It does not describe any scheme for Doppler measurement, which is part of the area of interest addressed by this invention.
U.S. Pat. No. 4,179,696 of Quesinberry contains a Kalman estimator which essentially transforms the line-of-sight (LOS) measurement data (i.e. range, angle, etc.) to obtain the target position as referenced to a stable coordinate system. This again is a quite different problem. It involves no scheme for a Doppler measurement and is essentially concerned with the transformation of one set of data (LOS coordinates) into a different set (fixed or stable coordinates) as is the case with the Virnot patent mentioned above.
The patent of Wu (U.S. Pat. No. 4,471,357) describes a method of two-dimensional processing of SAR images which is an entirely different problem than that of primary concern herein.
The following general commentary may be helpful in placing a proper perspective on the present invention as will be described hereinafter in relation to least squares algorithms and the Kalman filter. The basic idea of minimizing the sum of squares of the residuals goes back to Gauss in the 19th century who applied it to the study of problems in celestial mechanics. Since then, as is evidenced by a very large body of published scientific research, this generic idea has permeated through many scientific, engineering, and other fields. In this regard, the celebrated Kalman filter is also a specific implementation of the least squares algorithm; but, limited to a very structured problem of linear systems with Gaussian noise. For the more general nonlinear case (as is true with the environment of principal concern herein) there is no single scheme of a general nature. One possible solution, although not always satisfactory, is to use a linear approximation and apply the Kalman algorithm to this linearized model. A sinusoidal function can hardly be considered to be linear except over a very small segment. Thus, the above-referenced algorithms, although using least squares and Kalman filters, do not solve this problem.
Indeed, the fact that there are so many different techniques to solve the problem indicates the importance of the problem. This, however, also implies that there is no single technique superior to all others in all possible situations and/or with respect to different criteria such as computational complexity, statistical efficiency, etc.